Exercise 26.33 of PSLS 3e ------------------------- Measurement of goldfish ventilation rates for goldfish that had been acclimated to either a cold (12 degrees Celsius) or a warm (22 degrees Celsius) environment, when exposed to one of a range of acute test temperatures (10, 12, 15, 22 and 25 degrees Celsius). Each of the 10 treatment groups included 18 goldfish, for a total of 180 subjects. The design can be viewed as either a completely randomized design or a randomized block design, whereby the long-term acclimation would be a blocking factor. In the completely design, each of the 180 goldfish would have been randomly allocated to one of the 10 treatment groups. The fish would then have undergone their adaptation to their selected environment, and at the time of acute response measurement each would then have been exposed to its selected acute temperature. Descriptive analysis: MTB > WOpen "H:\VHM\VHM801\Datasets\Minitab\Chapter 26\ex26_033.mtw". Retrieving worksheet from file: ‘H:\VHM\VHM801\Datasets\Minitab\Chapter 26\ex26_033.mtw’ Worksheet was saved on 26/01/2014 MTB > Table 'Temperature' 'Acclimation'; SUBC> Layout 1 1; SUBC> DMissing 'Temperature' 'Acclimation'; SUBC> Means 'Ventilation'; SUBC> StDev 'Ventilation'; SUBC> Counts. Tabulated statistics: Temperature, Acclimation Rows: Temperature Columns: Acclimation cold warm All 10 58.89 33.94 46.42 21.23 17.35 22.91 18 18 36 12 55.83 39.83 47.83 18.44 24.43 22.82 18 18 36 15 76.83 59.72 68.28 13.54 16.74 17.34 18 18 36 22 88.56 78.28 83.42 24.88 25.82 25.53 18 18 36 25 124.83 129.00 126.92 37.22 30.72 33.70 18 18 36 All 80.99 68.16 74.57 34.64 41.42 38.61 90 90 180 Cell Contents: Ventilation : Mean Ventilation : Standard deviation Count MTB > Boxplot ( 'Ventilation' ) * 'Temperature'; SUBC> Group 'Acclimation'; SUBC> IQRBox; SUBC> Outlier. MTB > PPlot 'Ventilation'; SUBC> Normal; SUBC> Symbol; SUBC> FitD; SUBC> Grid 2; SUBC> Grid 1; SUBC> MGrid 1; SUBC> Panel 'Temperature' 'Acclimation'. Probability Plot of Ventilation The P-values for the A-D normality test: all above 0.3, except for 0.079 for (25,cold) Comments for descriptive analysis: ---------------------------------- The 10 group means vary substantially, from 33.9 (10,warm) to 129.0 (25,warm). The standard deviations are too variable to meet the rule based on the ratio of the largest to the smallest: s_max/s_min = 37.2/13.5 = 2.8. All the within-group distributions look reasonably regular, and only one of them gives a P-value for test of normality close to 0.05. So maybe apart from some concern with the standard deviations, the model assumptions seem to be met reaonable well. Statistical model: The model for two-way ANOVA can be written: vent_ijk = mu + alpha_i + beta_j + gamma_ij + eps_ijk where the errors (eps_ijk) are i.i.d. from N(0,sigma), and the indices i and j correspond to the row (temperature) and column (acclimation) factors, with I=5 and J=2. Statistical analysis (using brief Minitab menu for a start): MTB > Name c5 "RESI1" MTB > ANOVA 'Ventilation' = Temperature Acclimation Temperature* & CONT> Acclimation; SUBC> Residuals 'RESI1'; SUBC> GFourpack. ANOVA: Ventilation versus Temperature, Acclimation Factor Type Levels Values Temperature fixed 5 10, 12, 15, 22, 25 Acclimation fixed 2 cold, warm Analysis of Variance for Ventilation Source DF SS MS F P Temperature 4 157158 39289 68.13 0.000 Acclimation 1 7411 7411 12.85 0.000 Temperature*Acclimation 4 4235 1059 1.84 0.124 Error 170 98036 577 Total 179 266840 S = 24.0143 R-Sq = 63.26% R-Sq(adj) = 61.32% Residual Plots for Ventilation MTB > Interact 'Temperature' 'Acclimation'; SUBC> Response 'Ventilation'; SUBC> Full. Interaction Plot for Ventilation Comments for two-way ANOVA analysis: ------------------------------------ The ANOVA table shows a non-significant interaction (P=0.12), although the P-value is not too far from significance. The interaction plot shows a clear interaction pattern whereby the difference between the acclimation groups diminishes as the temperature increases. If such an interaction is of biological interest, it may be relevant to base model conclusions on the model with this interaction included, despite its non-significance. The main effects for acclimation and temperature are both strongly significant (P<0.0005). The ventilation generally increases with temperature and is larger in the cold acclimation group, except for at 25 degrees. The residual plots shows one rather extreme residual corresponding to an observation in the (25,cold) group (it would be of interest to compute the corresponding standardized residual, but that would require the model to be refit in the General Linear Model menu). Otherwise the distribution of the standardized residuals appears to be close to normal. The residual plot shows a slight fanning to the right. The two 25 degrees groups are more variable than the rest, something that could also be seen in the descriptive statistics above. It is not clear how serious this violation of model assumptions is. Both tests for equal variance (in the ANOVA menu, output not shown) give significant P-values, but the practical implication of this is not clear. Statistical analysis for square-transformed outcome (as requested): MTB > Name C4 'rootvent' MTB > Let 'rootvent' = sqrt('Ventilation') MTB > GLM; SUBC> Response 'rootvent'; SUBC> Nodefault; SUBC> Categorical 'Temperature' 'Acclimation'; SUBC> Terms Temperature Acclimation Temperature*Acclimation; SUBC> TMethod; SUBC> TAnova; SUBC> TSummary; SUBC> TCoefficients; SUBC> TEquation; SUBC> TFactor; SUBC> TMeans; SUBC> TDiagnostics 0; SUBC> Rtype 2; SUBC> GFOURPACK. General Linear Model: rootvent versus Temperature, Acclimation Method Factor coding (-1, 0, +1) Factor Information Factor Type Levels Values Temperature Fixed 5 10, 12, 15, 22, 25 Acclimation Fixed 2 cold, warm Analysis of Variance Source DF Adj SS Adj MS F-Value P-Value Temperature 4 516.32 129.079 61.87 0.000 Acclimation 1 41.28 41.281 19.79 0.000 Temperature*Acclimation 4 23.63 5.909 2.83 0.026 Error 170 354.68 2.086 Total 179 935.91 Model Summary S R-sq R-sq(adj) R-sq(pred) 1.44441 62.10% 60.10% 57.51% Coefficients Term Coef SE Coef T-Value P-Value VIF Constant 8.329 0.108 77.36 0.000 Temperature 10 -1.741 0.215 -8.09 0.000 1.60 12 -1.653 0.215 -7.68 0.000 1.60 15 -0.135 0.215 -0.63 0.531 1.60 22 0.686 0.215 3.19 0.002 1.60 Acclimation cold 0.479 0.108 4.45 0.000 1.00 Temperature*Acclimation 10 cold 0.480 0.215 2.23 0.027 1.60 12 cold 0.220 0.215 1.02 0.309 1.60 15 cold 0.059 0.215 0.27 0.784 1.60 22 cold -0.172 0.215 -0.80 0.426 1.60 Regression Equation rootvent = 8.329 - 1.741 Temperature_10 - 1.653 Temperature_12 - 0.135 Temperature_15 + 0.686 Temperature_22 + 2.844 Temperature_25 + 0.479 Acclimation_cold - 0.479 Acclimation_warm + 0.480 Temperature*Acclimation_10 cold - 0.480 Temperature*Acclimation_10 warm + 0.220 Temperature*Acclimation_12 cold - 0.220 Temperature*Acclimation_12 warm + 0.059 Temperature*Acclimation_15 cold - 0.059 Temperature*Acclimation_15 warm - 0.172 Temperature*Acclimation_22 cold + 0.172 Temperature*Acclimation_22 warm - 0.587 Temperature*Acclimation_25 cold + 0.587 Temperature*Acclimation_25 warm Fits and Diagnostics for Unusual Observations Obs rootvent Fit Resid Std Resid 82 14.697 11.065 3.632 2.59 R 115 8.944 5.977 2.967 2.11 R 116 8.944 5.977 2.967 2.11 R 120 2.828 5.977 -3.149 -2.24 R 124 2.000 5.977 -3.977 -2.83 R 156 4.472 8.708 -4.236 -3.02 R R Large residual Means Fitted Term Mean SE Mean Temperature 10 6.588 0.241 12 6.676 0.241 15 8.194 0.241 22 9.015 0.241 25 11.173 0.241 Acclimation cold 8.808 0.152 warm 7.850 0.152 Temperature*Acclimation 10 cold 7.547 0.340 10 warm 5.629 0.340 12 cold 7.374 0.340 12 warm 5.977 0.340 15 cold 8.732 0.340 15 warm 7.656 0.340 22 cold 9.322 0.340 22 warm 8.708 0.340 25 cold 11.065 0.340 25 warm 11.281 0.340 Residual Plots for rootvent MTB > Interact 'Temperature' 'Acclimation'; SUBC> Response 'rootvent'; SUBC> Full. Interaction Plot for rootvent Comments for second two-way ANOVA analysis: ------------------------------------------- Two major conclusions stand out: the model assumptions seem to be met better after transformation, and the interaction is now significant. In the residual plot, the variation is more evenly distributed across the range of fitted values. The most extreme residuals are now in the left tail, and overall the normal distribution plot looks better. Unless we believe that the observation with the extreme residual should really be dropped (the descriptive statistics don't indicate the observation to be totally outside the range of the others), the analysis on transformed scale may be somewhat better and hence preferable. The significant interaction (P=0.026) may be surprising, but the P-value in the first analysis was not so far from significance, and more importantly the concept of interaction is scale-dependent. That is, additivity (i.e., no interaction) exists (perfectly) on one scale, but after non-linear transformation the additivity no longer holds (perfectly), the question of statistical significance being another issue. In any case, it entirely reasonable and not too uncommon to see a significant interaction one scale and a non-significant interaction on another scale. If we decided to use the analysis on square-root scale, we would base our conclusions on the interaction plot and suitable pairwise comparisons between group means. The listing above included means (with SE) for the interaction, and we repeate the table here with the backtransformed means included as well: Means Fitted Term Mean SE Mean Backtransformed Temperature*Acclimation 10 cold 7.547 0.340 56.95 10 warm 5.629 0.340 31.69 12 cold 7.374 0.340 54.38 12 warm 5.977 0.340 35.72 15 cold 8.732 0.340 76.24 15 warm 7.656 0.340 58.61 22 cold 9.322 0.340 86.90 22 warm 8.708 0.340 75.83 25 cold 11.065 0.340 122.43 25 warm 11.281 0.340 127.26 In order to carry out pairwise comparisons, we compute a LSD-value for unadjusted comparisons, using tstar=t_.975(170)=1.974: LSD(.95) = tstar*s*sqrt(2/18) = 1.974*1.444*sqrt(2/18) = 0.95. This means, any two of the above means are significantly different if they differ by more than 0.95. Within two combined factors it is often most natural to carry out only comparisons with one factor fixed. First, we compare the two acclimations within each temperature: it is seen that significance holds for temperatures 10, 12 and 15, but not for 22 and 25 degrees. At 10, 12 and 15 degrees, the fish from cold acclimation have higher ventilation. Next we compare temperatures within each acclimation: acclimation=cold: 10,12 < 15,22 <25 acclimation=warm: 10,12 < 15,22 <25 So we have the same pattern in the two acclimation groups. If one wants to adjust for multiple comparisons, the Bonferroni can be used with a reduced number of comparisons corresponding to the ones just carried out. The number of comparisons carried out was 5*1 + 2*(5*4/2)) = 25, less than the total number of comparisons among 10 groups (10*9/2=45). The adjusted tstar value corresponds to a tail probability of 0.025/25=0.001, and is found (using software) to be 3.139. Therefore, the adjusted LSD becomes 3.139*s*sqrt(2/18)=1.51. With this value, substantially less of the comparisons become significant; for example, only the acclimation comparison at temperature 10 degrees is significant.